One of the foundations of pure mathematics is proof: showing mathematically, elegantly and thoroughly that a statement is true or false. But some statements in mathematics are taken as universally true, even though we cannot prove them. These are called axioms. For example, “Things which are equal to the same thing are also equal to one another” (Euclid's common notions). For many years, mathematicians have tried to find proofs for certain axioms, turning them into non-axiomatic statements.

But in 1931, Kurt Gödel showed that it is possible to prove that a statement is unprovable, literally rocking the entire mathematical community. But what does this proof actually look like?

First, we need to know about formal theories. Formal theories are sets of symbols governed by axioms and rules.

Gödel found that if a formal theory has enough rules in it to do maths, then it has one of either of these problems:

1) The formal theory has rules that make contradictions
2) The formal theory is incomplete, which means there are rules within it that I can’t prove using its other rules

Gödel used specific symbols to describe formal theories called the Gödels Numbering System. and then wrote "The Gödel Sentence” (represented by “G”):

The string of symbols above (“G”) essentially states, "G cannot be proven true within this formal theory”.

If G can be proven false (by other rules in the formal theory), then G can be proven true within said formal theory, which contradicts the proof you wrote that G is false.

If G can be proven true (by other rules in the formal theory) then G cannot be proven true within said formal theory, making the formal theory incomplete.

This means that certain rules in a (complete) formal theory have no proof.

You might have actually encountered something similar in other self-referential statements. For example:

In another world, people can either be truth-tellers (who only tell the truth) or liars (who only tell lies). Someone in said world says “This statement is false”.

If the statement is false, then it’s false that the statement is false and the speaker is a truth-teller, but if the statement is true, then it’s true that the statement is false, and so the speaker is a liar.